Find the equation of the circle passing through the points:
(i) (5, 7), (8, 1) and (1, 3)
(ii) (1, 2), (3, −4) and (5, −6)
(iii) (5, −8), (−2, 9) and (2, 1)
(iv) (0, 0), (−2, 1) and (−3, 2)

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Solution

(i) Let the required circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . ...(1)
It passes through (5, 7), (8, 1) and (1, 3).

Substituting the coordinates of these points in equation (1):

$74 + 10 g + 14 f + c = 0$ ...(2)
$65 + 16 g + 2 f + c = 0$ ...(3)
$10 + 2 g + 6 f + c = 0$ ...(4)

Simplifying (2), (3) and (4):

$g = \frac{- 29}{6}, f = \frac{- 19}{6}, c = \frac{56}{3}$

Equation of the required circle:
$x^{2} + y^{2} - \frac{29 x}{3} - \frac{19 y}{3} + \frac{56}{3} = 0$
$\Rightarrow$ $3 (x^{2} + y^{2}) - 29 x - 19 y + 56 = 0$

(ii) Let the required circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . ...(1)

It passes through (1, 2), (3, −4) and (5, −6).

Substituting the coordinates of these points in equation (1):

$5 + 2 g + 4 f + c = 0$ ...(2)
$25 + 6 g - 8 f + c = 0$ ...(3)
$61 + 10 g - 12 f + c = 0$ ...(4)

Simplifying (2), (3) and (4):
$g = - 11, f = - 2, c = 25$

The equation of the required circle is $x^{2} + y^{2} - 22 x - 4 y + 25 = 0$ .

(iii) Let the required circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . ...(1)

It passes through (5, −8), (−2, 9) and (2, 1).

Substituting the coordinates of these points in equation (1):

$89 + 10 g - 16 f + c = 0$ ...(2)
$85 - 4 g + 18 f + c = 0$ ...(3)
$5 + 4 g + 2 f + c = 0$ ...(4)

Simplifying (2), (3) and (4):
$g = 58, f = 24, c = - 285$

The equation of the required circle is $x^{2} + y^{2} + 116 x + 48 y - 285 = 0$ .

(iv) Let the required circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . ...(1)
It passes through (0, 0), (−2, 1) and (−3, 2).

Substituting the coordinates of these points in equation (1):

$c = 0$ ...(2)
$5 - 4 g + 2 f + c = 0$ ...(3)
$13 - 6 g + 4 f + c = 0$ ...(4)

Simplifying (2), (3) and (4):

$g = \frac{- 3}{2}, f = \frac{- 11}{2}, c = 0$

The equation of the required circle is $x^{2} + y^{2} - 3 x - 11 y = 0$ .

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